We first discuss the original minimax optimization loss [1]}, [2]}, [3]} for learning the MIW \(\omega (s,a)\) .
When given an offline dataset \(\mathcal {D}_{\mathrm {env}}\) , the objective is
\(\textstyle \leavevmode \xbox {resizebox}{\XMLaddatt {width}{397.10687pt}\min _{\omega \in W}\max _{f \in \mathcal {F}}\left\lbrace (1-\gamma )\mathbb {E}_{\mu _0,\pi }[f(s,a)] +\mathbb {E}_{(s,a, s^\prime )\sim d_{\pi _b,\gamma }^{P^{*}}, a^\prime \sim \pi (\cdot \,|\,s^\prime )}\left[ \gamma \omega (s,a)f(s^\prime , a^\prime ) - \omega (s,a)f(s,a)\right]\right\rbrace ,}\) 
